Anisotropic materials have properties that are directionally dependent. The elastic behavior of such materials has been discussed extensively in S. G. Leknitskii, Theory of Elasticity of an Anisotropic Body (English Translation, Mir Publishers, Moscow, 1981) and in T. C. T. Ting, Anisotropic Elasticity (Oxford University Press, New York, 1996). One observation of these studies is that the directions of principal stress and strain do not necessarily coincide. Another observation is that when anisotropic elements in the form of rectangular parallelepipeds are deformed under stress, they are deformed into oblique parallelepipeds. In two dimensions, this means that under stress, a rectangle becomes an oblique, or non-right, parallelogram.
Thermal expansion principals can be formulated in terms of elastic theory. As such, thermal expansion can likewise cause shape changes. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Oxford University Press, New York 1959), discloses that a circular cylinder having its expansion axis along one of the principal axes becomes an elliptic cylinder upon heating. However, in most treatments, thermal expansion is treated as an isotropic phenomenon, although the possibility of anisotropic thermal expansion has been recognized.
Despite the body of theory, few practicable applications have been taught. In U.S. Pat. No. 3,068,162 (Roche), the design of slot and key systems for the design of nuclear moderator structures is described in which the minimization of tolerances is important. However, Roche only describes a system for assembling together two pieces having substantially identical anisotropic expansions in which the orthogonal principal axes of expansion of one piece coincide with the orthogonal principal axes of expansion of the other piece.